On the Yamaha, I think the choices are Min, Max, and Std, correct? That is slightly different than the original poster's Sony with Off, Std, Max but equally confusing. Because his includes 'Off' (no compression), then I assume Max means maximum compression.
In the case of the Yamaha, when they say 'use Max to preserve the greatest dynamic range' that means that Max is no compression. It refers to 'maximum dynamic range' NOT 'maximum compression'. Min would be the absolute minimum dynamic range and therefore the most compression and Std would be somewhere in between.
So to translate:
No compression = None (Onkyo), Off (Sony), Max (Yamaha)
little compresion = Low (Onkyo), Std (Sony), Std (Yamaha)
a lot of compression = High (Onkyo), Max (Song), Min (Yamaha)
If you play a DD track either from a disc or from digital cable, you should be able to tell which is which when you switch between them. The setting for the small amount of compression sometimes doesn't make a very drastic change, but the maximum compression is very noticeable.